1549380323-Statistical Mechanics Theory and Molecular Simulation

Hamiltonian formulation 19

=−

∑N

i=1

p^2 i

2 mi

−U(r 1 ,...,rN). (1.6.2)

The function−L ̃(r 1 ,...,rN,p 1 ,...,pN) is known as theHamiltonianH:

H(r 1 ,...,rN,p 1 ,...,pN) =

∑N

i=1

p^2 i

2 mi

+U(r 1 ,...,rN). (1.6.3)

The Hamiltonian is simply the total energy of the system expressed as a function of
positions and momenta and is related to the Lagrangian by

H(r 1 ,...,rN,p 1 ,...,pN) =

∑N

i=1

pi·r ̇i(pi)−L(r 1 ,...,rN,r ̇ 1 (p 1 ),....,r ̇N(pN)). (1.6.4)

The momenta given in eqn. (1.6.1) are referred to asconjugateto the positions
r 1 ,...,rN.
The relations derived above also hold for a set of generalized coordinates. The
momentap 1 ,...,p 3 Nconjugate to a set of generalized coordinatesq 1 ,...,q 3 Nare given
by

pα=

∂L

∂q ̇α

, (1.6.5)

and the Hamiltonian becomes

H(q 1 ,...,q 3 N,p 1 ,...,p 3 N) =

∑^3 N

α=1

pαq ̇α(p 1 ,...,p 3 N)

−L(q 1 ,...,q 3 N,q ̇ 1 (p 1 ,...,p 3 N),...,q ̇ 3 N(p 1 ,...,p 3 N)). (1.6.6)

Now, according to eqn. (1.4.18), sinceGαβis a symmetric matrix, the generalized
conjugate momenta are

pα=

∑^3 N

β=1

Gαβ(q 1 ,...,q 3 N) ̇qβ. (1.6.7)

Inverting this, we obtain the generalized velocities as

q ̇α=

∑^3 N

β=1

G−αβ^1 (q 1 ,...,q 3 N)pβ, (1.6.8)

where the inverse of the mass-metric tensor is

G−αβ^1 (q 1 ,...,q 3 N) =

∑N

i=1

1

mi

(

∂qα

∂ri

)

·

(

∂qβ

∂ri

)

. (1.6.9)

It follows that the Hamiltonian in terms of a set of generalized coordinates is

H(q 1 ,...,q 3 N,p 1 ,...,p 3 N) =

1

2

∑^3 N

α=1

∑^3 N

β=1

pαG−αβ^1 (q 1 ,...,q 3 N)pβ